Analyzing Functions Graphically

Key Concepts

• Identify the common features of a function when given an equation or graph.
• Analyze domain and range of the function.
• Analyze maximum and minimum values of the function.
• Understand axes of symmetry of the function.
• Analyze end behaviors of the graphs.

Graph of the functions and its features

Q 1: Plot the points of each function on a graph. Explain the features of the function that represents the graph:  

Solution:   

Graph A:  

Common features of the graph A:  

Domain: [-2, 2]  

Range: [1, 5] 

There is no x-intercept.  

There is no y-intercept. 

Graph B: 

Common features of the graph B:   

Domain: [-2, 2]  

Range: [-1, 11] 

x-intercept: 1.7 

y-intercept: 5 

Graph C:   

Common features of the graph C:   

Domain: [-2, 2]  

Range: [3, 21] 

There is no x-intercept.  

y-intercept: 5 

Analyze Domain and Range   

The domain of a function is the set of all values for which the function is defined.  

The range of the function is the set of all values that the function takes.  

Example 1:  

The graph of the function: p(x)=|x|−1.

Domain: all real numbers 

To find the range of p(x): 

|x| ≥ 0

|x|−1≥−1

p(x)≥−1

Range:

y≥−1

Example 2:  

The graph of the function: q(x)=−x²+3

.  

Domain: all real numbers 

To find the range of q(x) : 

x²≥0

−x²≤0

−x²+3≤3

q(x)≤3

Range: y≤3. 

Analyze Maximum and Minimum Values 

Example 3:  

The graph of the function: f(x)=−2x+5.

Solution:  

The graph of the linear function f(x)=−2x+5 decreases at a constant rate.   So, there is no maximum or minimum value.  

Example 4:  

The graph of the function:

g(x)=−2x+3.

Solution:  

The graph of the function g(x)=−2x+3 is a translation of an exponential function.  

It is bounded above the asymptote y=3 which means that g(x)<3.  

It has no maximum because it is approaching 3 but never reaches 3.  

The function g also has no minimum. As x increases, g(x) decreases. 

Example 5:  

The graph of the function: h(x)=|x−2|−1.

Solution:  

The graph of the function h(x)=|x−2|−1 is a translation of an absolute value function.  

It opens upward so the function has a minimum value of – 1 at the vertex (2,−1). 

Understand Axes of Symmetry   

Example 6:  

The graph of the function: p(x)=5−|x+1|. 

Solution:   

Translations of the absolute value function always have an axis of symmetry passing through the vertex.  

Here the function

p(x)=5−|x+1| has an axis of symmetry x=−1 passing through the vertex (−1, 5). 

Example 7:  

The graph of the function: q(x)=(x+3).

Solution:  

Quadratic functions always have a vertical axis of symmetry. 

Here the quadratic function q(x)=(x+3)² has an axis of symmetry x=−3. 

Example 8:  

The graph of the function:

r(x)=√x+2

Solution:  

The function r(x)=√x+2 does not have an axis of symmetry.  

There is no way to fold the graph so that one side aligns with the other. 

Analyze End Behaviors of Graphs

Example 9:  

The graph of the function:

h(x)=x²−2x+1.

Solution:   

As x→∞ , the values of h(x) increases without bound.  

So, h(x)→∞.  

As x→−∞ , the values of h(x) also increases without bound.  

So, h(x)→∞ .    

Example 10:  

The graph of the function:

g(x)=∛x−2.

Solution:   

As x→∞ , the values of g(x) grow less and less steeply, but they do not approach to any asymptote. 

So, g(x)→∞.  

As x→−∞ , the values of g(x) also decreases.  

So, g(x)→−∞.   

Exercise

1. Sketch the graph of the function fx=√x-4 and identify its domain and range.
2. Use the graph of the function fx=5|x|-8 to identify its maximum and minimum value if they exist.
3. Describe the end behavior of the function fx=-7^x.

Concept Summary

What we have learned

• Identify the common features of a function when given an equation or graph.
• Analyze domain and range of the function.
• Analyze maximum and minimum values of the function.
• Understand axes of symmetry of the function.
• Analyze end behaviors of the graphs.